superconductivity near the mott transition a cluster dynamical mean field theory (cdmft) perspective...

Superconductivity near the Mott transition a Superconductivity near the Mott transition a Cluster Dynamical Mean Field Theory Cluster Dynamical Mean Field Theory

(CDMFT) perspective(CDMFT) perspective Gabriel KotliarGabriel Kotliar

Rutgers

Coherence and incoherence in stronly correlated systems. Coherence and incoherence in stronly correlated systems.

July 3-7 Rome Italy July 3-7 Rome Italy

Collaborators : G. Biroli M . Capone M Civelli K. Collaborators : G. Biroli M . Capone M Civelli K. Haule O. Parcollet T.D. Stanescu V. Kancharla Haule O. Parcollet T.D. Stanescu V. Kancharla

A.M.Tremblay B. Kyung D. Senechal A. GeorgesA.M.Tremblay B. Kyung D. Senechal A. Georges

References CollaboratorsReferences Collaborators• M. Capone and GK PRB 74, 54513(2006)• M. Civelli M. Capone A. Georges K. Haule O Parcollet T. Stanescu and GK cond-mat

0704.1486• M. Civelli et. al. PRL 95 106402(2005)• B. Kyung S. Kancharla D. Senechal A. Ms Tremblay M. CIvellli and GK PRB 73 165114(20060.• K. Haule and GK ( preprint)

Closely related work:Closely related work:

M. Capone M. Fabrizio C. Castellani and E. M. Capone M. Fabrizio C. Castellani and E. Tosatti Phys. Rev. Lett 93, 047001(2004) . Tosatti Phys. Rev. Lett 93, 047001(2004) . Science

296 2364 (2002).

CupratesCuprates

Damascelli, Shen, Hussain, RMP 75, 473 (2003)

Kappa Organics Kappa Organics

Phase diagram of (X=Cu[N(CN)2]Cl)S. Lefebvre et al. PRL 85, 5420 (2000), P. Limelette, et al. PRL 91 (2003)

F. Kagawa, K. Miyagawa, + K. Kanoda

PRB 69 (2004) +Nature 436 (2005)

PerspectivePerspective

U/t

t’/t

Doping Driven Mott Doping Driven Mott TransitionTransition

Pressure Driven Pressure Driven Mott transtionMott transtion

t-J Hamiltonian RVB P.W. Anderson (1987)t-J Hamiltonian RVB P.W. Anderson (1987)

Slave Boson Formulation: Baskaran Zhou Anderson (1987) Slave Boson Formulation: Baskaran Zhou Anderson (1987) Ruckenstein Hirschfeld and Appell (1987) Ruckenstein Hirschfeld and Appell (1987)

bb++i bi +fi bi +f++i fi fi = 1i = 1

Other RVB states with d wave symmetry. Flux phase or s+id G. Kotliar (1988) Affleck and Marston (1988) . Spectrum of excitation have point zeros like a a d –wave superconductor.

Superexchange Mechanism: proximity to Superexchange Mechanism: proximity to the Mott transition renormalizes down the Mott transition renormalizes down

kinetic energy, but not the superexchange.kinetic energy, but not the superexchange.•

Coherent Quasiparticles

Re

0

0b

Slave Boson Mean Field Theory Phase Diagram.

Formation of Singlets

Problems with the approach.Problems with the approach.

• Stability of the MFT. Ex. Neel order. Slave boson MFT with Neel order predicts AF AND SC. [Inui et.al. 1988] Giamarchi and L’huillier (1987).

• Gauge fluctuations destablize the mean field [Ubbens and Lee]

• Temperature dependence of the penetration depth [Wen and Lee , Ioffe and Millis ] . Theory:[T]=x-T x2 , Exp: [T]= x-T

• Z= x . Mean field is too uniform on the Fermi surface, in contradiction with ARPES.

• No proper description the incoherent regime and the coherent-incoherent and the incoherent regime.

Dynamical Mean Field TheoryDynamical Mean Field Theory• Map lattice model into quantum impurity problem in

a self consistent medium. • The quantum impurity problem is used to generate

local quantities, i.e. a local self energy.• From local quantities one reconstruct k dependent

spectral functions, susceptibilities, etc.• Single site, DMFT k independent self energy

(cumulant).]• Cluster extensions, incorporate additional k

dpendence.• Follow different mean field states, AF, normal,

supeconductor, etc as a function of parameters.

CLUSTER EXTENSIONS: umbiased reduction of the many body CLUSTER EXTENSIONS: umbiased reduction of the many body problem to a plaquette in a medium. problem to a plaquette in a medium.

11 23

24

( , ) (cos cos )

cos coslatt k kx ky

kx ky

wS =S +S +

+S

Reviews: Reviews: Georges et.al. RMP(1996). Th. Maier, M. Jarrell, Th.Pruschke, M.H. Hettler RMP (2005); G. Kotliar S. Savrasov K. Haule O. Parcollet V. Udovenko and C. Marianetti RMP (2006) . Employ different impurity solvers. ED (Civelli Capone) CTQMC (Haule)NCA (Haule)

Single site DMFT Qualitative Phase diagram of Single site DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillinga frustrated Hubbard model at integer filling

T/W Synthesis:Synthesis:

Brinkman RiceBrinkman Rice

HubbardHubbard

Castellani, C., C. DiCastro, D. Feinberg, and J. Ranninger,

1979, Phys. Rev. Lett. 43, 1957.

• Good description of the evolution of the

spectra and transport, not too close to the Mott transition, at relatively high temperatures. For example V2O3 ( Rozenberg et. al. 1996) K-organics (Limelette et.al. 2002).

• At lower temperatures, closer to the Mott transition, cluster description is necessary.

• Study at low temperatures the doping driven Mott transition.

The approach validates many crucial The approach validates many crucial features of the RVB theory. features of the RVB theory.

Tunnelling DOS (NCA-tJ): Gap (distance Tunnelling DOS (NCA-tJ): Gap (distance between coherence peaks) increases with between coherence peaks) increases with

decreasing doping. decreasing doping.

Order Parameter and Superconducting Gap do not scale for Order Parameter and Superconducting Gap do not scale for large U ! ED study in the SC state Capone and GK PRB large U ! ED study in the SC state Capone and GK PRB

(2006) Kancharla et. al. cond-mat 0508205.(2006) Kancharla et. al. cond-mat 0508205.

CDMFT on a plaquette gives rise to CDMFT on a plaquette gives rise to a “Dynamical RVB “pictures which a “Dynamical RVB “pictures which retains all the good features of the retains all the good features of the previous slave boson mft treatmentprevious slave boson mft treatment

• The quasiparticle residue, decreases with doping but the effective mass (Fermi velocity) remains finite. [M. Grilli and GK] PRL (1990)

• The gap in the tunneling density of states increases with decreasing doping.

• The ph asymmetry grows with the approach to the Mott insulator.

• Superconducting order parameter does not scale with the gap.

But with substantial two But with substantial two differences!!! which have important differences!!! which have important

consequencesconsequences

a) nodal antinodal dichotomya) nodal antinodal dichotomyb) vb) v decreses with decreasing decreses with decreasing

doping in superconductor. [Two-doping in superconductor. [Two-gap picture]gap picture]

Nodal Antinodal Dichotomy and pseudogap. T. Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK PRB (2006)Stanescu and GK PRB (2006)

Nodal Antinodal Dichotomy [Civelli et. al. Nodal Antinodal Dichotomy [Civelli et. al. (2007)](2007)]

Follow the “Follow the “normal state”normal state” with doping. with doping. Civelli et.al. PRL 95, Civelli et.al. PRL 95, 106402 (2005)106402 (2005)

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k U=16 t, 0) vs k U=16 t,

t’=-.3t’=-.3

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to Ek = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

K.M. Shen et.al. 2004

2X2 CDMFT

Doping Driven Mott transiton at low temperature, in 2d Doping Driven Mott transiton at low temperature, in 2d ((U=16 t=1, t’=-.3U=16 t=1, t’=-.3 ) Hubbard model ) Hubbard model

Spectral Function A(k,Spectral Function A(k,ω→ω→0)= -1/0)= -1/ππ G(k, G(k, ωω →→0) vs k0) vs kK.M. Shen et.al. 2004

2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)

Scaling of the || velocity in the Scaling of the || velocity in the superconductor with doping. superconductor with doping.

M. Civelli et. al. cond-mat M. Civelli et. al. cond-mat

K. Haule and GK> K. Haule and GK>

Consequences for linear term coefficient of the penetration depth. . K. Consequences for linear term coefficient of the penetration depth. . K. Haule and GKHaule and GK

Experiments:two superconducing gaps, with Experiments:two superconducing gaps, with opposite dependence on doping ? opposite dependence on doping ? Antinodal gap increases towards the Mott Antinodal gap increases towards the Mott insulator while vinsulator while v decreases? decreases?

• Coherence and single-particle excitations in the high-temperature superconductors. Guy Deutscher ,Nature 397, 410-412 (1999) Andreev reflection.

• M. Opel et. al. PRB 61, 9752 (2000) Venturini, F. et al., Doping dependence of the electronic Raman spectra in Phys. Chem. Solids, 63, 2345 (2001). Raman scattering.

LeTacon et. al. Two Energy Scales and two Quasiparticle Dynamics in the Superconducting . Nature Physics 2, 537 (2006) Raman scattering.

. . K. Tanaka, et. al Distinct Fermi-Momentum Dependent K. Tanaka, et. al Distinct Fermi-Momentum Dependent Energy Gaps in Deeply Underdoped Bi2212 . arXiv:cond-Energy Gaps in Deeply Underdoped Bi2212 . arXiv:cond-mat/0612048 .mat/0612048 . ARPES ARPES

M. C. Boyer et. al. arXiv:0705.1731 . Imaging the Two M. C. Boyer et. al. arXiv:0705.1731 . Imaging the Two Gaps of the High-TC Superconductor Pb-Bi2Sr2CuO6+x Gaps of the High-TC Superconductor Pb-Bi2Sr2CuO6+x Tunnelling. Tunnelling.

arXiv:0705.0111 Spectroscopic distinction between the arXiv:0705.0111 Spectroscopic distinction between the normal state pseudogap and the superconducting gap of normal state pseudogap and the superconducting gap of cuprate high T_{c} superconductors Li Yu, et. al. .cuprate high T_{c} superconductors Li Yu, et. al. . C- C- Axis Optical Spectrsocopy. Axis Optical Spectrsocopy.

• Metodological advantages. We can follow well defined phases as a function of parameters , doping temperature.

• Well defined (meta) stable states, in contrast to the old slave boson MFT approach.

• CDMFT treats properly the incoherent state, with short ranged magnetic correlations.

AF and superconductivity: M. Capone and GK PRB 74,054513AF and superconductivity: M. Capone and GK PRB 74,054513

AFM blue dashed line with circles and

dSC red solid line with squares order parameters as a function of

doping for four values of the repulsion U/ t=4,8,12, and 16. The

dSC order parameter is multiplied by a factor of 10 for graphical

purposes.

• Can we continue the superconducting state towards the Mott insulating state ?

For U > ~ 8t YES.

For U ~ < 8t NO, magnetism really gets in the way.

Evolution of the q integrated staggered Evolution of the q integrated staggered spin susceptilibty K. Haule and GK spin susceptilibty K. Haule and GK

(2006)(2006)

Conclusions: CDMFT studies of Conclusions: CDMFT studies of superconductivity near a Mott insulator. superconductivity near a Mott insulator.

Captures the essential RVB physics of the interplay of the Mott transition and superconductivity. Kinetic energy supression. Retains the good aspects of the slave boson MFT.

• Solves many problems of the earlier slave boson . [e.g.doping dependence of T linear term in the penetration depth ]

• Allows the continuation of spin liquid states as metastable states. Functional of local spectral functions.

• Nodal Antinodal dichotomy, emerges naturally.• Work in progress. No full solution of the CDMFT

eqs.and its lattice interpretation, (on the same level of single site DMFT), is available yet.

Happy Birthday Carlo!!!!

Temperature dependence of the arcs. Temperature dependence of the arcs. doping=.09 (underdoped) Plaquette doping=.09 (underdoped) Plaquette

DMFT. K. Haule and GKDMFT. K. Haule and GK

Lines of Zeros and Spectral Shapes. Lines of Zeros and Spectral Shapes. Stanescu and GK cond-matt 0508302Stanescu and GK cond-matt 0508302

Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K Interpretation in terms of lines of zeros and lines of poles of G T.D. Stanescu and G.K cond-matt 0508302cond-matt 0508302

Finite temperature view of the phase Finite temperature view of the phase diagram :optimal doping in the t-J diagram :optimal doping in the t-J model.K. Haule and GK (2006) model.K. Haule and GK (2006)

On the accuracy of CDMFTOn the accuracy of CDMFT

U/t=4.

Two Site Cellular DMFTTwo Site Cellular DMFT (G.. Kotliar et.al. PRL (2001)) in the 1D in the 1D Hubbard modelHubbard model M.Capone M.Civelli V. Kancharla C.Castellani and GK PRB

69,195105 (2004)T. D Stanescu and GK PRB (2006)

2424

On the interpretation of CDMFTOn the interpretation of CDMFT



2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)Senechal et.al Senechal et.al PRL94 (2005)PRL94 (2005)

RVB phase diagram of the Cuprate RVB phase diagram of the Cuprate Superconductors. Superexchange.Superconductors. Superexchange.

• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• Approach the Mott insulator , Z, charge stiffness , TBE=Tcoh goes to zero. M* finite.

• Superconducting dome. Pseudogap evolves continously into the superconducting state.

G. Kotliar and J. Liu Phys.Rev. B 38,5412 (1988)

Related approach using wave functions:T. M. Rice group. Zhang et. al. Supercond Scie Tech 1, 36 (1998, Gross Joynt and Rice (1986) M. Randeria

N. Trivedi , A. Paramenkanti PRL 87, 217002 (2001)



2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)Senechal et.al Senechal et.al PRL94 (2005)PRL94 (2005)

Pseudoparticle picturePseudoparticle picture

How is the Mott insulatorHow is the Mott insulatorapproached from the approached from the

superconducting state ?superconducting state ?

Work in collaboration with M. Capone M Civelli O Parcollet

Nodal Antinodal Dichotomy and pseudogap. T. Nodal Antinodal Dichotomy and pseudogap. T. Stanescu and GK cond-matt 0508302Stanescu and GK cond-matt 0508302

Superconducting DOS Superconducting DOS

=.08

= .16

Superconductivity is destroyed by transfer of spectral weight. M. Capone et. al. Similar to slave bosons d wave RVB.

Superconductivity in the Hubbard model Superconductivity in the Hubbard model role of the role of the Mott transitionMott transition and and influence of the super-exchangeinfluence of the super-exchange. .

( work with M. Capone et.al V. Kancharla.et.al ( work with M. Capone et.al V. Kancharla.et.al CDMFT+ED, 4+ 8 sites t’=0) . CDMFT+ED, 4+ 8 sites t’=0) .

cond-mat/0508205cond-mat/0508205 Anomalous superconductivity in doped Anomalous superconductivity in doped Mott insulator:Mott insulator:Order Parameter and Superconducting Gap . Order Parameter and Superconducting Gap . They scale together for small U, but not for large U. S. They scale together for small U, but not for large U. S. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G. Kancharla M. Civelli M. Capone B. Kyung D. Senechal G.

Kotliar andA.Tremblay. Cond mat Kotliar andA.Tremblay. Cond mat 05082050508205 M. Capone M. Capone (2006). (2006).

M. Capone and GK cond-mat 0511334 . Competition fo M. Capone and GK cond-mat 0511334 . Competition fo superconductivity and antiferromagnetism. superconductivity and antiferromagnetism.

Superconducting DOS Superconducting DOS

=.08

= .16

Superconductivity is destroyed by transfer of spectral weight.. Similar to slave bosons d wave RVB. M. Capone et. al

Anomalous Self EnergyAnomalous Self Energy. (from Capone et.al.) Notice the . (from Capone et.al.) Notice the remarkable increase with decreasing doping! True remarkable increase with decreasing doping! True

superconducting pairing!! U=8tsuperconducting pairing!! U=8t

Significant Difference with Migdal-Eliashberg.

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT within plaquette Cellular DMFT

• Rich Structure of the normal state and the interplay of the ordered phases.

• Work needed to reach the same level of understanding of the single site DMFT solution.

• A) Either that we will understand some qualitative aspects found in the experiment. In which case the next step LDA+CDMFT or GW+CDMFT could be then be used make realistic modelling of the various spectroscopies.

• B) Or we do not, in which case other degrees of freedom, or inhomogeneities or long wavelength non Gaussian modes are essential as many authors have surmised.

• Too early to tell, talk presented some evidence for A.

.

OutlineOutline

• Introduction. Mott physics and high temperature superconductivity. Early Ideas: slave boson mean field theory. Successes and Difficulties.

• Dynamical Mean Field Theory approach and its cluster extensions.

• Results for optical conductivity.

• Anomalous superconductivity and normal state.

• Future directions.

Temperature dependence of the spectral Temperature dependence of the spectral weight of CDMFT in normal state. Carbone weight of CDMFT in normal state. Carbone

et al, see also ortholani for CDMFT. et al, see also ortholani for CDMFT.

Larger frustration: t’=.9t U=16tLarger frustration: t’=.9t U=16tn=.69 .92 .96n=.69 .92 .96

M. Civelli M. CaponeO. Parcollet and GK M. Civelli M. CaponeO. Parcollet and GK

PRL (20050PRL (20050

. Spectral weight integrated up to 1 eV of the three BSCCO . Spectral weight integrated up to 1 eV of the three BSCCO films. a) under-films. a) under-

doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼doped, Tc=70 K; b) optimally doped, Tc=80 K; c) ∼overdoped, Tc=63 K; the fulloverdoped, Tc=63 K; the full

symbols are above Tc (integration from 0+), the open symbols symbols are above Tc (integration from 0+), the open symbols below Tc, (integrationfrom 0, including th weight of the below Tc, (integrationfrom 0, including th weight of the

superfuid).superfuid).

H.J.A. Molegraaf et al., Science 295, 2239 (2002). A.F. Santander-Syro et al., Europhys. Lett. 62, 568 (2003). Cond-mat 0111539. G. Deutscher et. A. Santander-Syro and N. Bontemps. PRB 72, 092504(2005) . Recent review:

• P.W. Anderson. Connection between high Tc and Mott physics. Science 235, 1196 (1987)

• Connection between the anomalous normal state of a doped Mott insulator and high Tc. t-J limit.

• Slave boson approach. <b> coherence order parameter. singlet formation order parameters.Baskaran Zhou Anderson , (1987)Ruckenstein Hirshfeld and Appell (1987) .Uniform Solutions. S-wave superconductors. Uniform RVB states.

Other RVB states with d wave symmetry. Flux phase or s+id ( G. Kotliar (1988) Affleck and Marston (1988) . Spectrum of excitation have point zerosUpon doping they become a d –wave superconductor. (Kotliar and Liu 1988). .

The simplest model of high Tc’s

t-J, PW Anderson

Hubbard-Stratonovich ->(to keep some out-of-cluster quantum fluctuations)

BK Functional, Exact

cluster in k space cluster in real space

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

DMFT Qualitative Phase diagram of a DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillingfrustrated Hubbard model at integer filling

T/W

Georges et.al. Georges et.al. RMP (1996) RMP (1996)

Kotliar Kotliar Vollhardt Vollhardt

Physics Today Physics Today (2004)(2004)

Single site DMFT and kappa organics. Qualitative phase Single site DMFT and kappa organics. Qualitative phase

diagram Coherence incoherence crosoverdiagram Coherence incoherence crosover. .

Dependence on periodization scheme. Dependence on periodization scheme.

Energetics and phase separation. Right Energetics and phase separation. Right U=16t Left U=8t U=16t Left U=8t

t’=0

Phase diagram

Temperature Depencence of Integrated spectral weight

Pseudoparticle picturePseudoparticle picture

Optical Conductivity near optimal Optical Conductivity near optimal doping. [DCA ED+NCA study, K. doping. [DCA ED+NCA study, K.

Haule and GK]Haule and GK]

Behavior of the Behavior of the optical mass and the optical mass and the plasma frequency.plasma frequency.

Magnetic SusceptibilityMagnetic Susceptibility

References and CollaboratorsReferences and Collaborators

• References:• M. Capone et. al. in preparation• M. Capone and G. Kotliar cond-mat cond-mat/0603227 • Kristjan Haule, Gabriel Kotliar cond-mat/0605149• M. Capone and G.K cond-mat/0603227• Kristjan Haule, Gabriel Kotliar cond-mat/0601478

• Tudor D. Stanescu and Gabriel Kotliar cond-mat/0508302• S. S. Kancharla, M. Civelli, M. Capone, B. Kyung, D.

Senechal, G. Kotliar, A.-M.S. Tremblay cond-mat/0508205• M. Civelli M. Capone S. S. Kancharla O. Parcollet and G.

Kotliar Phys. Rev. Lett. 95, 106402 (2005)

P. W. Anderson, P. W. Anderson, ScienceScience 235235, 1196 (1987) , 1196 (1987)

RVB phase diagram of the Cuprate RVB phase diagram of the Cuprate Superconductors. Superconductors.




RVB Approach Anderson (1987)RVB Approach Anderson (1987)

• Understand the physics resulting from the proximity to a Mott insulator in the context of the simplest models. [ Leave out disorder, electronic structure,phonons …]

• Follow different “states” as a function of parameters. [Second step compare free energies which will depend more on the detailed modelling…..]

• Solve the plaquette mean field equations!!!! Work in progress.

Phys Rev. B 72, 092504 (2005)

cluster-DMFT, cond-mat/0601478

Kinetic energy change in t-J K Haule and GK

Kinetic energy decreases

Kinetic energy increases

Kinetic energy increases

Exchange energy decreases and gives

largest contribution to condensation energy

cond-mat/0503073

.. • AFunctional of the cluster Greens function. Allows the investigation of the normal

state underlying the superconducting state, by forcing a symmetric Weiss function, we can follow the normal state near the Mott transition.

• Earlier studies use QMC (Katsnelson and Lichtenstein, (1998) M Hettler et. T. Maier et. al. (2000) . ) used QMC as an impurity solver and DCA as cluster scheme. (Limits U to less than 8t )

• Use exact diag ( Krauth Caffarel 1995 ) as a solver to reach larger U’s and smaller Temperature and CDMFT as the mean field scheme. • Recently (K. Haule and GK ) the region near the superconducting –normal state

transition temperature near optimal doping was studied using NCA + DCA-CDMFT .• DYNAMICAL GENERALIZATION OF SLAVE BOSON ANZATS -(k,)+= /b2 -(+b2 t) (cos kx + cos ky)/b2 + • b--------> b(k), ----- (), k• Extends the functional form of self energy to finite T and higher frequency.• Larger clusters can be studied with VCPT CPT [Senechal and Tremblay, Arrigoni,

Hanke ]

CDMFT study of cuprates


• The approach to the Mott insulator renormalizes the kinetic energy Trvb increases.

• The proximity to the Mott insulator reduce the charge stiffness , TBE goes to zero.

• Superconducting dome. Pseudogap evolves continuously into the superconducting state.




Copper oxide superconducors CuOCopper oxide superconducors CuO2 2

Kappa organics Kappa organics

Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003)

H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998)

t’/t ~ 0.6 - 1.1

Photoemission spectra near Photoemission spectra near the antinodal direction in a the antinodal direction in a

Bi2212 underdoped sample. Bi2212 underdoped sample. Campuzano et.alCampuzano et.al

EDC along different parts of the zone, from Zhou et.al.

Origin of the ph asymmetryOrigin of the ph asymmetry

Problems with the approach.Problems with the approach.

• Stability of the MFT. Ex. Neel order. Slave boson MFT with Neel order predicts AF AND SC. [Inui et.al. 1988] Giamarchi and L’huillier (1987).

Copper oxide superconducors CuOCopper oxide superconducors CuO2 2

Kappa organics Kappa organics

Y. Shimizu, et al. Phys. Rev. Lett. 91, 107001(2003)

H. Kino + H. Fukuyama, J. Phys. Soc. Jpn 65 2158 (1996), R.H. McKenzie, Comments Condens Mat Phys. 18, 309 (1998)

t’/t ~ 0.6 - 1.1

Ut

t’t’’

H ij t i,j c i c j c j

c i Uinini

Model Hamiltonians Model Hamiltonians

Photoemission spectra near Photoemission spectra near the antinodal direction in a the antinodal direction in a

Bi2212 underdoped sample. Bi2212 underdoped sample. Campuzano et.alCampuzano et.al


• Proximity to Mott insulator renormalizes the kinetic energy Trvb increases.

• Proximity to the Mott insulator reduce the charge stiffness, and QPcoherence scale . T BE goes to zero.

• Superconducting dome.• Pseudogap with d wave

symmetry.




ApproachApproach

• Understand the physics resulting from the proximity in the context of the simplest models.

• Leave out disorder, electronic structure,phonons,inhomogeneous structures.

• Follow different “states” as a function of parameters.

• Second step compare free energies which will depend more on the detailed modelling

• Local (plaquette ) Mott physics. Leave out long wavelength collective modes.

• Look at experiments.• Work in progress. The framework and the resulting

CDMFT equations are very non trivial to solve.

Lower Temperature, AF and SCLower Temperature, AF and SCM. Capone and GK, M. Capone and GK,

AF

AF+SC

SC

AFSC

Mott Phenomeman and High Temperature Superconductivity Mott Phenomeman and High Temperature Superconductivity Began Study of minimal model of a doped Mott insulator Began Study of minimal model of a doped Mott insulator

within plaquette Cellular DMFT within plaquette Cellular DMFT

• Rich Structure of the normal state and the interplay of the ordered phases.

• Work needed to reach the same level of understanding of the single site DMFT solution.

• A) Either that we will understand some qualitative aspects found in the experiment. In which case the next step LDA+CDMFT or GW+CDMFT could be then be used make realistic modelling of the various spectroscopies.

• B) Or we do not, in which case other degrees of freedom, or inhomogeneities or long wavelength non Gaussian modes are essential as many authors have surmised.

• Too early to tell, talk presented some evidence for A.

.

Dynamical Mean Field Theory. Cavity Construction.Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992).A. Georges and G. Kotliar PRB 45, 6479 (1992).

†

0 0 0

( )[ ( ' ] ( '))o o o oc c U n nb b b

s st m tt

t t ¯

¶+ D-

¶- +òò ò

,ij i j i

i j i

J S S h S- -å å eMF offhH S=-† †

, ,

( )( )ij ij i j j i i ii j i

t c c c c U n n

*

( )V Va a

a a

ww e

D =-å

† † † † †Anderson Imp 0 0 0 0 0 0 0

, , ,

( +c.c). H c A A A c c UcV c c c

A(A())

1010

Finite T, DMFT and the Energy Landscape Finite T, DMFT and the Energy Landscape of Correlated Materials of Correlated Materials

T

DMFT Qualitative Phase diagram of a DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer fillingfrustrated Hubbard model at integer filling

T/W Synthesis:Synthesis:

Brinkman RiceBrinkman Rice

HubbardHubbard

Castellani et.al.Castellani et.al.

Kotliar RuckensteinKotliar Ruckenstein

FujimoriFujimori

Single site DMFT and kappa organics. Qualitative phase Single site DMFT and kappa organics. Qualitative phase

diagram Coherence incoherence crosoverdiagram Coherence incoherence crosover. .

Finite T Mott tranisiton in CDMFT Finite T Mott tranisiton in CDMFT O. Parcollet O. Parcollet

G. Biroli and GK PRL, 92, 226402. (2004))G. Biroli and GK PRL, 92, 226402. (2004))

CDMFT results Kyung et.al. (2006)CDMFT results Kyung et.al. (2006)

Evolution of the spectral function Evolution of the spectral function at low frequency.at low frequency.

( 0, )vs k A k

If the k dependence of the self energy is weak, we expect to see contour lines corresponding to t(k) = const and a height increasing as we approach the Fermi surface.

k

k2 2

k

Ek=t(k)+Re ( , 0)

= Im ( , 0)

( , 0)Ek

k

k

A k

Evolution of the k resolved Spectral Evolution of the k resolved Spectral Function at zero frequency. (Function at zero frequency. (Parcollet Biroli and GK Parcollet Biroli and GK

PRL, 92, 226402. (2004)) )PRL, 92, 226402. (2004)) ) ( 0, )vs k A k

Uc=2.35+-.05, Tc/D=1/44. Tmott~.01 W

U/D=2 U/D=2.25



2X2 CDMFT

Nodal Region

Antinodal Region

Civelli et.al. PRL 95 (2005)Civelli et.al. PRL 95 (2005)

Larger frustration: t’=.9t U=16tLarger frustration: t’=.9t U=16tn=.69 .92 .96n=.69 .92 .96

M. Civelli M. CaponeO. Parcollet and GK M. Civelli M. CaponeO. Parcollet and GK

PRL (20050PRL (20050

superconductivity near the mott transition a cluster dynamical mean field theory (cdmft) perspective...

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